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Linking number of modular knots

2023/01/03 by James Rickards, Rickards, James
Mathematics · #11F23 #37C27 #37E15 (Primary) 11E16 #57K10 #57K31 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.01334

openalex publication_date 2023/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the linking number of two modular knots in the space PSL(2, ℤ)\backslashPSL(2, ℝ) with the trefoil filled in, which answers a question posed by Ghys in 2007. This computation is realized through the correspondence between modular links and Lorenz links, and can be thought of as an intersection number involving Conway topographs. We compare this to a second formula for the linking number of Lorenz links, which was proven by Stephen F. Kennedy in 1994.

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